Problem: Simplify the following expression: $ z = \dfrac{-5}{2} + \dfrac{-5k + 2}{-k + 10} $
Solution: In order to add expressions, they must have a common denominator. Multiply the first expression by $\dfrac{-k + 10}{-k + 10}$ $ \dfrac{-5}{2} \times \dfrac{-k + 10}{-k + 10} = \dfrac{5k - 50}{-2k + 20} $ Multiply the second expression by $\dfrac{2}{2}$ $ \dfrac{-5k + 2}{-k + 10} \times \dfrac{2}{2} = \dfrac{-10k + 4}{-2k + 20} $ Therefore $ z = \dfrac{5k - 50}{-2k + 20} + \dfrac{-10k + 4}{-2k + 20} $ Now the expressions have the same denominator we can simply add the numerators: $z = \dfrac{5k - 50 - 10k + 4}{-2k + 20} $ $z = \dfrac{-5k - 46}{-2k + 20}$ Simplify the expression by dividing the numerator and denominator by -1: $z = \dfrac{5k + 46}{2k - 20}$